The advent of nanosatellites has given rise to a new challenge in the field of space propulsion. These satellites, typically weighing between 1 and 10 kg, require very minute forces for executing maneuvers for station keeping, orbital corrections, and attitude control. These forces are often of the order of a few micronewtons or less. Although the design for thrusters capable of delivering these forces has been around for some time [1], the need exists for a simple, low cost measurement method that resolves these sub-micro-Newton forces accurately and reliably.
Thrust stands, which use the principle of a torsion balance, were conceived by Charles Coulomb to measure the electrostatic force leading to the discovery of the Coulomb Law [2], and later used by Henry Cavendish to measure the gravitational force between two masses [3]. A thrust stand is essentially a torsion spring, which undergoes angular deflections under the action of a torque. This angular deflection, generated by an unknown force acting at a known distance, can be measured as linear displacement of the balance arm at a known distance from the pivot.
Jamison et al [4], Zeimer [5], Gamero-Castaño et al [5], and Gamero-Castaño [7] and Yang et al [8], have all built thrust stands with the same fundamental working principle, albeit with different methods of calibration, damping techniques, and displacement measurement. Table I summarizes the details and the steady-state force resolution achieved by each of these.
TABLE IReview of thrust stands with sub-microNewton level resolutionLowestThrustCalibrationDampingDisplacementResolutionMeasuredTeamSourceMechanismMeasurement(μN)(μN)Jamison et al [4]OrificeViscous Oil BathLVDT<10.088Thruster(±2-16%)Ziemer [5]ImpactDamping coilLVDT<11.0 (±20%)PendulumGamero-CastañoElectrostaticElectrostaticFiber Optic LDS0.010.11et al [6]Gamero-CastañoElectrostaticElectrostaticFiber Optic LDS0.037.89[7]Yang et al [8]FreeAir dampingAutocollimator0.09NotOscillationsmentioned
Note that the orifice thruster used by Jamison et al [4] was calibrated using DSMC techniques, and the lowest thrust measured was 88.8 nN. Actual steady state thrust measured by Ziemer [5] was 1 μN using a FEEP thruster. The lowest thrust measured by Gamero-Castaño et al [6] was 0.11 μN using an electrospray source, and 7.89 μN using a colloid thruster by Gamero-Castaño [7]. Yang et al [8] used a pendulum stage suspended from 502 μm titanium fiber which acts as the torsion spring. No damping other than that provided by surrounding air was employed, and the thrust stand was calibrated by measuring the moment of inertia of the setup and oscillation frequency. The resolution for steady-state thrust is mentioned to be 0.09 μN. A mechanism for damping of torsion balance oscillations was demonstrated by Polzin et al [10]. However, Polzin's thrust stand had an ultimate resolution of 50 μN,
As reported in Table I, several methods of calibration have been reported in literature. Many of these rely on a device that can produce an accurately known force against which the system is calibrated, with exceptions like the impact pendulum used by Ziemer [5] that relies on applying a known impulse instead of a steady-state force to back out the system dynamics. The orifice thruster, used by Jamison et al [4], is calibrated using Monte Carlo methods, which introduce a certain amount of uncertainty (±12.6%) in the exact force that the thruster delivers. The electrostatic force calibration implemented by Gamero-Castaño et al [6] and Gamero-Castaño [7] has its own disadvantages. The following error analysis shows that this electrostatic method inherently introduces multiple sources of error, necessitating a better calibration method.
The force between two identical parallel plate electrodes separated by a distance L and having area A is given by
                    F        =                              1            2                    ⁢                                    ɛ              ⁡                              (                                  V                  L                                )                                      2                    ⁢          A                                    (        5        )            Combining Eq. 4 and 5, the calibration constant kθ can be written as,
                              k          θ                =                              ɛ            2                    ·                      V            2                    ·                      1                          L              2                                ·          A          ·                      l            2                    ·                      1            x                                              (        6        )            
Applying error propagation analysis to this equation yields the following equation for relative uncertainty in kθ,
                                          ∂                          k              θ                                            k            θ                          =                                                            (                                                      2                    ⁢                    dV                                    V                                )                            2                        +                                          (                                                      2                    ⁢                    d                    ⁢                                                                                  ⁢                    L                                    L                                )                            2                        +                                          (                                  dA                  A                                )                            2                        +                                          (                                                      2                    ⁢                    dl                                    l                                )                            2                        +                                          (                                  dx                  x                                )                            2                                                          (        7        )            
As can be seen, there are five sources of error in the above calibration method, some of which can be controlled more precisely than others. The applied voltage V, electrode area A, and the moment length l can be controlled to a fair degree of accuracy. However, the electrode gap L (typically 1 mm), and the sensor resolution dx (5 nm as per manufacturer specifications at www.philtec.com, LDS Model D100) present the biggest challenge in controlling the spread of the calibration data. Equation 7 suggests that a 10% error in L alone leads to a 20% error in kθ. Accounting for errors from the other sources would increase this figure further.